Quasi-nilpotency of Generalized Volterra Operators on Sequence Spaces

Abstract

We study the quasi-nilpotency of generalized Volterra operators on spaces of power series with Taylor coefficients in weighted ell ^p spaces 1<p<+infty . Our main result is that when an analytic symbol g is a multiplier for a weighted ell ^p space, then the corresponding generalized Volterra operator T_g is bounded on the same space and quasi-nilpotent, i.e. its spectrum is {0}. This improves a previous result of A. Limani and B. Malman in the case of sequence spaces. Also combined with known results about multipliers of ell ^p spaces we give non trivial examples of bounded quasi-nilpotent generalized Volterra operators on ell ^p. We approach the problem by introducing what we call Schur multipliers for lower triangular matrices and we construct a family of Schur multipliers for lower triangular matrices on ell ^p, 1<p<infty related to summability kernels. To demonstrate the power of our results we also find a new class of Schur multipliers for Hankel operators on ell ^2 , extending a result of E. Ricard.